Sedimentary rocks may be carbon-dated by inference if they contain carbonaceous fossils.

The same may be true for igneous rocks, but finding suitable carbon samples may be unlikely.

They would assume that today's utensil is older than the one from 1000 years ago, on the basis of handiwork.

So is the part about the carbon dating of nearby organic material true?

Radiometric dating has been used to determine the ages of the Earth, Moon, meteorites, ages of fossils, including early man, timing of glaciations, ages of mineral deposits, recurrence rates of earthquakes and volcanic eruptions, the history of reversals of Earth's magnetic field, and many of other geological events and processes.